Sum of n Terms
The sum of the first n terms of an AP is S_n = n/2 [2a + (n − 1)d]. The version CAT students reach for far more often is S_n = n/2 (first + last), because in most word problems you already know (or can quickly find) both end terms. Both come from the same insight Gauss famously used: pair the first term with the last, the second with the second-last, and every pair sums to (first + last); there are n/2 such pairs. A third lens is the average: since the terms are evenly spaced, their mean is just (first + last)/2, so S_n = n × average. Keep two standard sums on instant recall: 1 + 2 + … + n = n(n + 1)/2 and the sum of the first n odd numbers = n². These let you evaluate sums of multiples or evenly spaced lists without ever writing the full series.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Term & general form
| nth term | a_n = a + (n − 1)d |
|---|---|
| Common difference | d = a_n − a_(n−1) |
| Number of terms | n = (last − first)/d + 1 |
| mth from the end | l − (m − 1)d (l = last term) |
| Neighbour-average property | a_n = (a_(n−1) + a_(n+1)) / 2 |
Sum & shortcuts
| Sum of n terms | S_n = n/2 [2a + (n − 1)d] |
|---|---|
| Sum via first & last | S_n = n/2 (first + last) |
| Sum from average | S_n = n × (average term) |
| First n naturals | 1 + 2 + … + n = n(n + 1)/2 |
| First n odd numbers | 1 + 3 + … + (2n − 1) = n² |